Classification of 3-dimensional Bihom-Associative and Bihom-Bialgebras

BiHom-associative and BiHom-Hopf algebras (BiHom-bialgebras with antipode structure) have many applications in various areas of mathematics and physics. Based on a significant and growing topic, the current paper aims to investigate the structure of (non-)unital BiHom-associative algebras and explore the algebraic varieties of BiHom-associative (bi-)algebras up to dimension 3. We use the correlations between the structural constants to provide the desired classification results. These results further enable us to differentiate between each isomorphism type of the 3-dimensional BiHom-associative and BiHom-bialgebras inside some equivalence classes. Finally based on our findings, we classified BiHom-Hopf algebras up to dimension 3. These results are useful for understanding related algebraic structures and present a substantial advancement.